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Generalized semilattices and m-degrees of index sets. (English. Russian original) Zbl 0708.06005

Algebra Logic 28, No. 5, 369-378 (1989); translation from Algebra Logika 28, No. 5, 555-569 (1989).
A generalized (upper) semilattice is a poset P supplied with a family of binary operations satisfying certain conditions. A subset \(I\subset P\) is said to be an ideal of a generalized semilattice P if I is a lower subset in the poset P closed under all binary operations. It is proved that every ideal I determines a congruence relation \(\theta\) (I) such that the quotient algebra P/\(\theta\) (I) is a semilattice. In addition, P/\(\theta\) (I) is a distributive semilattice provided that P is a distributive generalized semilattice. The property of distributivity is fulfilled, for example, in every generalized semilattice of m-degrees of index sets.
Reviewer: V.N.Salij

MSC:

06A12 Semilattices
03D30 Other degrees and reducibilities in computability and recursion theory
03D45 Theory of numerations, effectively presented structures
03D35 Undecidability and degrees of sets of sentences
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References:

[1] V. L. Selivanov, ”The structure of degrees of unsolvability of index sets,” Algebra Logika,18, No. 4, 463–480 (1979). · Zbl 0439.03025
[2] T. M. Kuz’mina, ”The structure ofm -degrees of unsolvability of index sets of families of partial recursive functions,” Algebra Logika,20, No. 1, 55–68 (1981).
[3] V. L. Selivanov, ”The structure of generalized index sets,” Algebra Logika,21, No. 4, 472–491 (1982).
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